Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2856 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14757 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,10)(5,26)(6,13)(7,12)(8,14)(9,18)(11,27)(16,34)(17,46)(19,33)(20,32)(21,50)(22,52)(23,40)(24,36)(25,37)(28,38)(29,39)(30,44)(31,48)(35,54)(41,45)(42,47)(43,56)(49,57)(51,58)(53,60)(55,59)(61,62)(63,64);(1,3,6,18,8,19,7,20)(2,9,12,15,14,32,13,33)(4,21,23,16,25,44,24,45)(5,17,27,47,28,48,29,49)(10,34,36,50,37,41,40,30)(11,31,26,42,39,46,38,57)(22,51,53,43,55,62,54,63)(35,58,59,64,60,61,52,56);(1,4)(2,10)(3,16,19,45)(5,22)(6,24)(7,23)(8,25)(9,30,32,50)(11,35)(12,40)(13,36)(14,37)(15,41,33,34)(17,43,48,63)(18,21,20,44)(26,52)(27,54)(28,55)(29,53)(31,56,46,64)(38,59)(39,60)(42,61,57,58)(47,51,49,62);(1,5)(2,11)(3,17)(4,22)(6,27)(7,29)(8,28)(9,31)(10,35)(12,26)(13,38)(14,39)(15,42)(16,43)(18,47)(19,48)(20,49)(21,51)(23,53)(24,54)(25,55)(30,56)(32,46)(33,57)(34,58)(36,59)(37,60)(40,52)(41,61)(44,62)(45,63)(50,64)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305145);